Exercises: Representation on the number line

Look at the following number line and identify which integer corresponds to each of the letters marked with a dot:

Draw a number line in your notebook and represent the following integers on it: $+4$, $-3$, $0$, $+1$ and $-5$. Use a distance of $1$ in. between each number.

On a thermometer, temperatures above zero are considered positive and below zero, negative. Represent the following temperatures from a winter week on the vertical line of the thermometer: $-2 ^\circ C$, $+5 ^\circ C$ and $0 ^\circ C$.

Indicate which integer is the opposite of the one represented by the orange dot on the line. Also represent that opposite on the same line.

A diver is at a depth of $-10$ feet (below sea level). If he ascends $4$ feet, in what position is he now? Represent this on a vertical number line where $0$ is sea level.

Look at the line and write the missing integers in the empty boxes, knowing that the distance between each mark is $1$ unit.

Remember that on the number line, the further to the right a number is, the larger it is. Based on this, use the symbols $>$ (greater than) or $<$ (less than) to compare these pairs of numbers:

1. a) $-4$ _____ $+2$
2. b) $-1$ _____ $-5$
3. c) $+3$ _____ $0$
4. d) $-2$ _____ $+1$

Write the integers whose modulus is equal to $3$. Then, represent them on a number line and explain what you observe regarding their position relative to zero.

Look at the following number line on which three points have been marked, $A$, $B$ and $C$. Taking into account the position of the number $0$ and $+2$, identify which integer each letter represents.

Draw a number line and represent on it all the integers $x$ that simultaneously satisfy these two conditions:

1. Their modulus is less than $5$, that is: $|x| < 5$.

2. They are numbers strictly less than $-1$.

A thermal inspection drone makes the following movements on the vertical axis of an industrial chimney. If we consider the mouth of the chimney to be at point $0$ feet:

• It starts at a height of $+12$ ft.

• It descends $15$ feet to check a crack.

• It goes back up $6$ feet to take a reading.

Represent the route on a vertical line and mark the drone's final position with a highlighted colored dot. At what height is it relative to the mouth of the chimney?

On the number line, the midpoint between two integers is the place that is exactly the same distance from both.

• Represent the numbers $-6$ and $+2$.

• Visually find the midpoint between them and state what number it represents.

• What is the distance (in units) from that midpoint to either of the two ends?

Represent the following integers on a number line. Choose a suitable scale (where each mark on the line can be worth more than 1 unit) so that they all fit in your drawing:

$S = \{ -15, +5, -20, +25, 0, -5 \}$

Identify which integers are represented by points $M$ and $N$ on this line, knowing that the distance between two consecutive marks is always the same.

Three scientists are studying an underwater volcano. Their positions relative to sea level ($0$ ft) are:

• Researcher A: Is in a submarine at $-450$ feet.

• Researcher B: Is at the top of the volcano, at $+150$ feet above sea level.

• Researcher C: Is on a boat at $0$ feet.

Represent these three positions on a vertical line. If researcher A wants to go up to researcher B's position, how many feet must he ascend in total? Explain how you see this on the line.

Draw a number line and place the number $-3$ on it. Then, represent:

• Point $A$, which is the opposite of $-3$.

• Point $B$, which is $5$ units to the left of $-3$.

• Point $C$, which is $2$ units to the right of $A$.

What number does point $C$ represent? Which is the smallest number of the four you have marked on the line?

An explorer is looking for a buried treasure on a straight path that we represent as a number line. The map gives him the following clues:

1. The treasure is in a position represented by an integer $x$.

2. The distance between $x$ and the number $-2$ is exactly $5$ units.

3. The absolute value of $x$ is greater than $4$.

Represent the possible locations of the treasure on the number line and determine at which exact number or numbers it could be found.

A surveillance drone moves along a horizontal axis. Its charging base is at point $0$. It makes the following consecutive movements:

• It starts at the base and moves to point $+7$.

• From there, it moves back $12$ units.

• Finally, it moves to a position $P$ such that the total distance traveled from the start is $25$ units.

If the drone always moves in a straight line, at which integer on the number line is point $P$? Is there more than one possible solution? Represent this graphically.

We are designing a special thermometer for a physics experiment. We must mark four critical points: $A$, $B$, $C$ and $D$, which satisfy these conditions:

• $A$ is the opposite of $+3$.

• $B$ has an absolute value of $5$ and is to the left of $A$.

• $C$ is an integer such that $-1 < C < 4$ and its distance to $A$ is the same as its distance to $D$.

• $D$ is the largest possible integer that is less than $+6$.

Represent the four points on the same number line and order them from smallest to largest absolute value.

Two cities, Villafría and Villacalor, are connected by a straight road that we imagine as a number line. Villafría is at point $-15$ and Villacalor is at point $+21$.

A cell tower is to be installed at a point $M$ that is exactly the same distance from both cities.

1. What integer represents the position of the cell tower $M$?

2. If a car leaves Villafría towards the cell tower and breaks down when it has traveled $\frac{1}{3}$ of the total distance between the city and the cell tower, at what number on the line has it stopped?

In a video game, a character moves along a line of squares numbered with integers. The player must place "traps" on all integers $n$ that simultaneously satisfy these two conditions:

• The modulus of the number is less than or equal to $6$: $|n| \le 6$.

• The distance between the number $n$ and the number $-3$ is greater than $4$ units.

Draw a number line and mark with an orange "X" all the positions where the player must place a trap. How many traps are there in total?

A historian is creating a digital timeline. He defines the year $0$ as the moment of a great discovery.

• Civilization Alfa emerged in the year $-450$ and lasted $300$ years.

• Civilization Beta began just as Alfa ended and lasted twice as long.

• Civilization Gamma is situated on the line so that its starting year is the midpoint between the start of Alfa and the end of Beta.

Represent the start and end points of each civilization on a number line (using a suitable scale, for example, 1 unit = 100 years). In what year did civilization Gamma begin?

Consider three integers $a, b$ and $c$ located on the number line of which we know the following:

• $a$ is the opposite of the modulus of $-8$.

• $c$ is a number such that its distance to $0$ is half the distance from $a$ to $0$, and $c$ is positive.

• $b$ is a number that is the same distance from $a$ as it is from $c$.

Draw the line, place the three points and calculate the value of the following expression using their positions:

$$|a| + |b| - |c|$$

Imagine that the number line is a taut rope. We tie a weight at the number $0$ and another at the number $-12$.

1. Find the position of a point $P$ (an integer) that is at a distance of $4$ units from point $-6$. How many solutions are there?

2. If we now look for a point $Q$ that satisfies that its distance to $+2$ is twice its distance to $-1$, what integers could $Q$ be? Identify them and mark them on the number line.